Simplify and expand the following expression: $ \dfrac{2p - 7}{p - 8}-\dfrac{3p}{p + 10} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(p - 8)(p + 10)$ Multiply the first term by $\dfrac{p + 10}{p + 10}$ $ \begin{align*} \dfrac{2p - 7}{p - 8} \times \dfrac{p + 10}{p + 10} & = \dfrac{(2p - 7)(p + 10)}{(p - 8)(p + 10)} \\ & = \dfrac{2p^2 + 13p - 70}{(p - 8)(p + 10)}\end{align*} $ Multiply the second term by $\dfrac{p - 8}{p - 8}$ $ \begin{align*} \dfrac{3p}{p + 10} \times \dfrac{p - 8}{p - 8} & = \dfrac{(3p)(p - 8)}{(p + 10)(p - 8)} \\ & = \dfrac{3p^2 - 24p}{(p + 10)(p - 8)}\end{align*} $ Now we have: $ = \dfrac{2p^2 + 13p - 70}{(p - 8)(p + 10)} - \dfrac{3p^2 - 24p}{(p + 10)(p - 8)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2p^2 + 13p - 70 - (3p^2 - 24p)}{(p - 8)(p + 10)} $ $ = \dfrac{2p^2 + 13p - 70 - 3p^2 + 24p}{(p - 8)(p + 10)} $ $ = \dfrac{-p^2 + 37p - 70}{(p - 8)(p + 10)}$ Expand the denominator: $ = \dfrac{-p^2 + 37p - 70}{p^2 + 2p - 80}$